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How can time-series data be declared as stationery?

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It is stationary when the variance and mean of the series are constant with time. 

Here is a visual example: 

stationery

In the first graph, the variance is constant with time. Here, X is the time factor and Y is the variable. The value of Y goes through the same points all the time; in other words, it is stationary.

In the second graph, the waves get bigger, which means it is non-stationary and the variance is changing with time.

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